Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters

Abstract

In this paper we study the asymptotics (as n → ∞) of the sequences of Laguerre polynomials with varying complex parameters α depending on the degree n. More precisely, we assume that α n = nA n , and lim n A n = A ∈ C. This study has been carried out previously only for α n ∈ R, but complex values of A introduce an asymmetry that makes the problem more difficult.The main ingredient of the asymptotic analysis is the right choice of the contour of orthogonality, which requires the analysis of the global structure of trajectories of an associated quadratic differential on the complex plane, which may have an independent interest.While the weak asymptotics is obtained by reduction to the theorem of GoncharRakhmanov-Stahl, the strong asymptotic results are derived via the non-commutative steepest descent analysis based on the Riemann-Hilbert characterization of the Laguerre polynomials.

Root-counting Measures of Jacobi Polynomials and Topological Types and Critical Geodesics of Related Quadratic Differentials

“…In many special cases statements similar to Proposition 6 can be found in the literature, see e.g. recent [1] and references therein. Proposition 6 allows us, under mild nondegeneracy assumptions, to formulate necessary and sufficient conditions for the existence of a motherbody measure for (4.1) which however are difficult to verify.…”

Trajectories of Quadratic Differentials for Jacobi Polynomials with Complex Parameters

“…Its proof is based on the so-called Teichmüller lemma (see [20,Theorem 14.1]) and follows literally the arguments that have been used in [1,Lemma 4]. We omit repeating them here for the sake of brevity.…”

Combinatorial description of jumps in spectral networks defined by quadratic differentials

“…Short trajectories of q turn to play an important role also in potential theory, approximation theory and other branches of mathematics. For example, short trajectories of rational quadratic differentials describe limiting distributions of certain types of orthogonal polynomials, see e.g., [14,15,16]. Motivated by applications to minimal surfaces, Bruce and O'Shea published a preprint [9], where the short trajectories characterized umbilical points and the geometry of unfolding.…”